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WindowTitle->Lecture 18 MIT 3.016 (Fall 2009)   \251 W. Craig Carter  \
2003--2009
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Cell["\<\
Fourier Theory, Fourier Transforms, Diffraction and Computational Simulation \
of Transmission Electron Microscopy\
\>", "Title",
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Cell["\<\
Acknowledgement: Prof. S. Allen graciously provided useful physical insight \
on diffraction interpretation in these notes.\
\>", "Text",
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Creating Sample Atomic Data and Lattices for Fourier Analysis and Diffaction \
Simulation.\
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Here we create an image such as might be seen in an transmission electron \
microscope. For our first set of  data, we use intererence of two sine waves \
to produce a simulation of the density of scattering centers in an atomic \
lattice.  Most of the physical aspects of atomic imaging and diffraction can \
be simulated with the two-dimensional techniques that are produced in these \
notes.\
\>", "Text",
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We show the simplest visulations of our data, we will improve on the coloring \
methods below.  Here we show our two dimensional crystal as a gray-intensity \
map and with a three dimensional plot of scattering intensity over an x-y \
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\>", "Text",
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Cell["\<\
Improving Visualization of Data and Diffraction with Contrast Functions\
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In order to view the data better, we wish to develop contrast functions. \
Experience suggests that it is useful to construct two type of contrast, this \
one, normalcontrast will be used as a ColorFunction when we want to resolve \
the entire range of intensities.\
\>", "Text",
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This second version, highcontrast, will be useful when we wish to resolve \
differences in the low-intensity region. Note how the colors change rapidly \
in the lower region.\
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Cell["Visualizing Atomic Densities", "Subtitle",
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Cell["\<\
ImagePlot will be our \"work-horse\" function to dispay images, as well as \
Fourier data.  It takes an rectangular array of data as input, and has thee \
optional arguments for a plot-label, the colorfunction to use (i.e., \
highcontrast or normalcontrast), and an indication of the image size.\
\>", "Text",
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Here we show the normal contrast and the highcontrast images of the simulated \
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Cell["Discrete Fourier Transforms on Simple 2D Lattices", "Subtitle"],

Cell["\<\
We create a function that shows the original data, its Fourier transform, and \
then its inverse transform (hopefully) back to the original image.\
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Cell["\<\
We will see the peaks of the image are located near (kx,ky) = 2 \[Pi] \
(a,b)/(Image Size), where 
(a,b) = {(0,0), (Image Size,0), (0,Image SIze), (Image Size, Image Size)}.  \
These correspond to the longest wavelength periodicities.\
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Cell["\<\
The magnitude of fourier the fourier transform corresponds to the \
constructive interference of wave - like objects scattered from a real - \
space set of data.  When the real-space data has translational symmetry, such \
as in a lattice, the fourier intensity results in bright spots corresponding \
to periodicities present in the lattice.The distance between spots is \
inversely proportional to the distance between periodic scattering features, \
and the orientation of the (kx, ky) vector corresponds to the normal to the \
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Cell["\<\
The middle image is a good simulation of what is observed for electron \
diffraction in the TEM.  The white spot (at the center) is called the \
\"direct beam\" or the \"transmitted beam\" and the other spots correspond to \
diffraction peaks from different (hkl) planes (or (hk) in 2D).  The pattern \
corresponds the the reciprocal lattice of the crystal object.  A vector from \
the origin (white spot) to a particular peak is the reciprocal lattice vector \
for that particular set of diffracting planes.\
\>", "Text",
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Here is an alternative visualiazation of the diffraction peaks using a third \
dimension to indicate the intensity of the peak.  We write it so we can \
specify the range optionally.\
\>", "Text",
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Look for a maximum in the atomic density map, and replace a circle around it \
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\>", "Text",
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We will show some examples of the diffraction and image reconstruction of \
different kinds of simulated data, these data will be explored below in the \
Computational Microscopy sections.\
\>", "Text",
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In the above, the defect causes diffuse intensity to appear in the pattern. \
Using methods produced below, we can investigate  how much of reciprocal \
space is needed to reproduce the defect in the inverse transform (i.e., the \
image of the scatters that is produced in a TEM). \
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Function to Create A Lattice of Squares : 
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First look at an example with no noise, we create a lattice of squares with \
lattice vectors (8,4) and (16,16).  This function will create smaller unit \
cells if the last vectors are divisors of the total size. Here we set the \
edge of the square to 1 and indicate that we want 0,0 noise in the x,y \
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We reconstruct the example from above, but introduce random unit \
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The appearance of diffuse intensity (e.g. the intensity that does not \
correspond to the bright spots for the ideal lattice) corresponds to random \
shifts in the original figure. This is a simulation of isotropic thermal \
vibrations. In crystals, the amplitudes would be larger in elastically soft \
directions.  It is instructive  to use the last two arguments in the \
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effect on the diffration pattern--the streaks that you will observe \
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voltage difference and sending them towards a target.  The electrons interact \
with the target and scatter. After the electrons have passed through the \
sample, they are focused with a magnetiic objective lens (or typically \
lenses). This lens produces a plane at which electrons scattered in the same \
direction arrive\nat the same point---this is the diffraction pattern.  \
Because diffraction transforms periodic elements into points, it is closely \
related to the fourier transform.  An image of the target is created beyond \
this diffraction plane.  An operator of an electron microscope can toggle \
between looking at the diffraction-plane or the image-plane.\n\nSpecific \
periodic features can be imaged by blocking all the electrons except those \
that appear at a point in the diffrations plance. To put it another way, one \
puts an aperature around a particular spot in order to selectively image the \
features that from that spot's periodicity. We can do this with a Fourier \
transform by selectling only a circular subset of points to be subsequently \
inversely transformed into an image.\n\nThe \"Bright-Field Image\" consists \
of using a central aperature around the direct beam to block off all others \
from contributing to the image.\n\nThe \"Dark-Field Image\" consists of \
selecting a specific diffracted peak with the aperature and using that to \
form an image\n\nA \"Structure image,\" or a \"lattice image,\" uses the \
direct beam and one or more diffracted beams to form the image.  In this \
case, the aperatures are typically much larger than for bright- or dark-field \
imaging. \n\nAperature size is effectively limited because of spherical \
aberrations that become significant for beams that are \"off-axis\" by a \
significant amount.  So, in practice one can only use ",
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Here is a simple example of four grains with the same modulated density, but \
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Diffraction Spot Microscopy on grain structure.  Notice that if you leave the \
small yellow aperature at the center, and move the green aperature over other \
diffraction spots, then the grain in that particular orientation will \"light \
up\" and its modulation will appear.  Now toggle to the \"Dark-Field \
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Diffraction Spot Microscopy on Idealized Lattice.  Notice that when you \
select pairs of spots, that particular atom planes are visualized. Use the \
purple aperature to select a spot on the same ray (i.e., with the same plane \
normals), the modulation doubles. You are visualizing (abc)-planes in one \
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Diffraction Spot Microscopy on Noisy Lattice.  Notice that when you select \
pairs of the brightest spots, you still get a fairly robust image of those \
planes, even though they are not perfectly alligned in the original image.  \
Most of the information is contained in the sharp peaks and those must \
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diffused over the entire Fourier spectrum.\
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Frivolous Example: Beautiful Data are in the Eye of the Beholder\
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Import some random intensities that we have collected off the web:\
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